L10: Real Intertemporal Model with Money: Key Concepts, Derivations & Policy Experiments

Exam Strategy and Linking Micro/Macro

• Always start from the equilibrium definitions that will be provided in the exam appendix (household FOCs, firm FOCs, market-clearing conditions, government budget constraint).
• Show BOTH perspectives:
  • Micro: indifference-curve or firm optimisation diagrams (e.g. leisure–consumption, leisure
    today–leisure
    tomorrow, investment schedule).
  • Macro: Aggregate‐Supply (YS) / Aggregate‐Demand (YD) diagram + money-market diagram.
• Explain the full logic chain: identify the exogenous shock → locate where it enters the equations → trace through labour market, production function, intertemporal choices, money market.
• Marks are awarded for coherent narrative, not for memorising a rigid template. Always write the economic intuition (marginal benefit = marginal cost stories) as well as the algebra/graphs.

Recap: Aggregate Supply & Demand Without Money

• YS curve
  • Combines labour-market equilibrium $MPL=W/P$ and firm production $Y=Zf(K,N)$.
• YD curve
  • Represents the (intertemporal) optimal choices of households (C, L) and firms (I) together with G.
  • Downward sloping in $(R,Y)$ space: a higher real interest rate raises the cost of current C & I.
• Last week’s model was entirely real; prices and money were absent.

Introducing Money

Why add money?

• Allows variables to be expressed in nominal terms; lets us analyse price level and inflation.
• Real world has multiple assets differing in liquidity; money is the most liquid.
• Model used: Cash-in-Advance (CIA) / liquidity-constraint approach.
  • Consumers & firms must hold money to purchase goods/investment.
  • Captures the idea that some transactions cannot be settled on credit alone.

Money as a Liquid Asset

• Benefits of holding money: immediate purchasing power; avoids transaction cost of credit.
• Costs: money earns zero nominal return, so the opportunity cost is the nominal interest rate $R$ obtainable on bonds (liquidity premium).

Extended Government Budget Constraint

$G<em>t + (1+R</em>{t-1})B<em>{t-1} = T</em>t + B<em>t + (M</em>t-M_{t-1})$

• Government + central bank consolidated: can finance spending via taxes, new debt, or money creation.

Consumer Side with CIA

• Timing within the period $t$:
  1. Begin with stocks $M<em>{t-1},\;B</em>{t-1}(1+R_{t-1})$.
  2. Pay nominal taxes $P<em>tT</em>t$; choose new bond position $B_t$.
  3. Must set aside cash $M<em>t^-$ to buy consumption $C</em>t$; may also spend via credit $X_t$.
• Cash-in-Advance / liquidity constraint
  $P<em>t C</em>t \le M<em>t^- + P</em>t X_t$
• End-of-period budget (real terms)
  $C<em>t + \frac{B</em>t}{P<em>t} + \frac{M</em>t}{P<em>t} = Y</em>t - X<em>t - h(X</em>t) + \frac{(1+R<em>{t-1})B</em>{t-1}}{P<em>t} + \frac{M</em>{t-1}}{P<em>t} - T</em>t$
• $h(\cdot)$: increasing, convex cost of using credit/debit facilities; crucial for an interior money-credit mix.

Firms

• Face an analogous liquidity constraint to pay wages/investment ⇒ also demand money.

Fisher Relation

• Arbitrage between money (return $0$) and bonds (nominal return $R$) while prices grow at rate $i$ (inflation):
  $1+r = \frac{1+R}{1+i} \;\;\Rightarrow\;\; r \approx R - i$
• $r$ real interest rate (appears in YS/YD diagrams).

Money Demand Derivation

• Marginal benefit of extra credit $= P<em>t(1+R)$; marginal cost $= h'(X</em>t)$.
• Equate to yield optimal $X^*$ ⇒ obtain household money demand.
• Aggregating consumers & firms: \frac{Mt}{Pt} = L(Yt, rt), \quad \frac{\partial L}{\partial Y}>0, \; \frac{\partial L}{\partial r}<0
  • Positive in income (scale effect), negative in real interest (opportunity cost).

Money Market Equilibrium Diagram

• Downward-sloping $L(\cdot)$ against $1/P$ (or $R$).
• Vertical supply at policy-chosen $M_t$.
• Intersection determines the price level $P_t$ (given real activity).

Classical Dichotomy & Neutrality

• Real allocation determined by real side (labour, production, intertemporal choices).
• Nominal variables adjust so $\text{Real money demand = Real money supply}$.
• With fully flexible prices, a **once-off change in nominal $M$ shifts $P$ one-for-one – no effect on $C,I,L,Y,r$ ⇒ *neutrality of money* / no money illusion.
• Keynesian models break the dichotomy by assuming sticky prices ⇒ short-run real effects.

Comparative-Static Experiments

1. Helicopter Drop (ΔM financed by equal lump-sum tax cut)

• $M$ up, $T$ down same amount ⇒ household present-value budget unchanged.
• Money-market: supply shifts right; $P$ jumps up proportionally so $M/P$ unchanged.
• All real variables (N, C, I, Y, r) unchanged ⇒ illustrates money neutrality.

2. Negative Productivity Shock (Z↓)

• Production function & $MPL$ shift down → labour demand ↓.
• YS shifts left; at initial YD real interest $r$ rises.
• Higher $r$ ⇒
  • Consumption & investment ↓ (substitution > income effect) ⇒ YD moves down along curve.
  • Labour supply ↑ (future returns higher) partly offsets N fall.
• Net: N ↓, Y ↓, r ↑.
• Higher $r$ raises opportunity cost of money ⇒ real money demand ↓ ⇒ $P$ rises.
• Consistent with observed counter-cyclicality of prices and pro-cyclicality of C, I, wages.

3. Financial-Sector Shock (Cost of Credit h↑)

• $h'(\cdot)$ up ⇒ households/firms substitute toward money, away from credit.
• Money-demand curve shifts right; with fixed $M$, price level $P$ must fall.
• Real side: extra credit cost enters expenditure identity ⇒ YD curve shifts left ⇒ output & employment fall (demand-driven recession).
• Real interest rate falls (liquidity preference effect); labour supply may shift via intertemporal substitution.
• Captures flight-to-liquidity episodes (e.g. 2008 crisis, COVID onset).

Business-Cycle Regularities Explained

• Model (with TFP shocks) predicts:
  • \text{Corr}(P, Y) < 0 (prices counter-cyclical).
  • \text{Corr}(C,Y),\;\text{Corr}(I,Y),\;\text{Corr}(N,Y),\;\text{Corr}(w,Y) > 0 (real aggregates pro-cyclical).
• Provides RBC-style micro foundation for observed correlations.

Policy Implications

• Short-run: with flexible prices, conventional monetary policy ineffective (neutrality).
• Long-run: persistent inflation distorts C–L choice, lowers employment → welfare costs; justifies inflation targeting.
• Central bank actions (QE, interest-rate targeting) operate via changing $M$; must conduct open-market operations to hit target.
• Sticky-price (Keynesian) extensions restore a short-run role for stabilisation policy.

Modelling Choices & Caveats

• CIA constraint is a shortcut; deeper models derive money demand from search & contracting frictions.
• Inside money (bank-created deposits) ignored; only outside money (central-bank issued) in constraint.
• Perfect foresight / certainty assumed for inflation; Fisher relation would use expected inflation in stochastic models.
• Strong assumption of lump-sum taxation drives the pure neutrality result; financing via debt or G would alter outcomes.

Useful Formulae & Notation Reference

• Fisher relation: $1+r = \dfrac{1+R}{1+i} \approx R - i$.
• Real money demand: \dfrac{M}{P}=L(Y,r),\; LY>0,\;Lr<0.
• Government constraint: $G<em>t + (1+R</em>{t-1})B<em>{t-1}=T</em>t+B<em>t+(M</em>t-M_{t-1})$.
• CIA (consumer): $P<em>t C</em>t \le M<em>t^- + P</em>t X<em>t$ with $h'(X</em>t)=P<em>t(1+R</em>t)$ at optimum.
• Production: $Y<em>t = Z</em>t f(K,N<em>t)$, $MPL=Z</em>t f<em>N(K,N</em>t)=W<em>t/P</em>t$.

Real-World Data Nuggets Mentioned

• Australia: physical cash now funds ≈ $13\%$ of household transactions (RBA).
• Historical hyperinflations (Argentina, Brazil 1980s) illustrate dangers of printing money for deficits.
• “Flight to quality” during 2007-09 GFC: spike in liquidity demand, safe-asset premium.

Checklist When Answering Similar Questions

1. Identify the exogenous shock & where it appears in equations.
2. Update micro first-order conditions if necessary (e.g. $h'(X)$, $MPL$, Fisher).
3. Shift the appropriate curve(s) in:
   • Labour market (ND, NS).
   • Production function diagram.
   • YS / YD diagram.
   • Money-market diagram.
4. Determine sequence of endogenous adjustments (r, N, Y, P, C, I, w).
5. Comment on business-cycle properties & policy relevance.
6. State any auxiliary assumptions (e.g. size of opposing effects) explicitly.

"Piece and trace out all these logical connections and then put it together as a coherent explanation" – Lecturer’s core instruction.